Confidence interval estimation for negative binomial group distribution

Negative binomial group distribution was proposed in the literature which was motivated by inverse sampling when considering group inspection: products are inspected group by group, and the number of non-conforming items of a group is recorded only until the inspection of the whole group is finished. The non-conforming probability p of the population is thus the parameter of interest. In this paper, the confidence interval construction for this parameter is investigated. The common normal approximation and exact method are applied. To overcome the drawbacks of these commonly used methods, a composite method that is based on the confidence intervals of the negative binomial distribution is proposed, which benefits from the relationship between negative binomial distribution and negative binomial group distribution. Simulation studies are carried out to examine the performances of our methods. A real data example is also presented to illustrate the application of our method.     

Properties and applications of the Poisson-reciprocal inverse Gaussian distribution

The barely known continuous reciprocal inverse Gaussian distribution is used in this paper to introduce the Poisson-reciprocal inverse Gaussian discrete distribution. Several of its most relevant statistical properties are examined, some of them directly inherited from the reciprocal of the inverse Gaussian distribution. Furthermore, a mixed Poisson regression model that uses the reciprocal inverse Gaussian as mixing distribution is presented. Parameters estimation in this regression model is performed via an EM type algorithm. In light of the numerical results displayed in the paper, the distributions introduced in this work are competitive with the classical negative binomial and Poisson-inverse Gaussian distributions.     

Improved approximations for the inverse moments of the positive binomial distribution

Several alternatives to the most common approximation to the inverse moments of the positive binomial distribution are obtained. The method is based on equating moments and gives considerably better approximations for some values of the parameters.     

Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.     

Comparison of Shared Frailty Models for Kidney Infection Data under Exponential Power Baseline Distribution

Shared frailty models are often used to model heterogeneity in survival analysis. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, four shared frailty models with frailty distribution gamma, inverse Gaussian, compound Poisson, and compound negative binomial with exponential power as baseline distribution are proposed. These models are fitted using Markov Chain Monte Carlo methods. These models are illustrated with a real life bivariate survival data set of McGilchrist and Aisbett (1991) related to kidney infection, and the best model is suggested for the data using different model comparison criteria.     

A family of bivariate binomial distributions generated by extreme bernoulli distributions

In this paper, bivariate binomial distributions generated by extreme bivariate Bernoulli distributions are obtained and studied. Representation of the bivariate binomial distribution generated by a convex combination of extreme bivariate Bernoulli distributions as a mixture of distributions in the class of bivariate binomial distribution generated by extreme bivariate Bernoulli distribution is obtained. A subfamily of bivariate binomial distributions exhibiting the property of positive and negative dependence is constructed. Some results on positive dependence notions as it relates to the bivariate binomial distribution generated by extreme bivariate Bernoulli distribution and a linear combination of such distributions are obtained.     

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.     

The two-sided power-binomial distribution for modeling binary count data

The two-sided power (TSP) distribution is a flexible two-parameter distribution having uniform, power function and triangular as sub-distributions, and it is a reasonable alternative to beta distribution in some cases. In this work, we introduce the TSP-binomial model which is defined as a mixture of binomial distributions, with the binomial parameter p having a TSP distribution. We study its distributional properties and demonstrate its use on some data. It is shown that the newly defined model is a useful candidate for overdispersed binomial data.     

On some distributions arising in inverse cluster sampling

In this paper, distributions of items sampled inversely in clusters are derived. In particular, negative binomial type of distributions are obtained and their properties are studied. A logarithmic series type of distribution is also defined as a limiting form of the obtained generalized negative binomial distribution.     

A generalization of the beta–binomial distribution

Summary.  The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution.     

A New Markov Binomial Distribution

In this article, we introduce a two-state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is related to the interrupted Markov chain. Some characterization properties of the geometric distributions are given. Recursion formulas and probability mass functions for the NB distribution and the new binomial distribution are derived.     

Attribute control chart for some popular distributions

The binomial distribution is often used to display attribute control data. In this paper, a statistical model is settled for attribute control chart under truncated life test. By Burr X & XII, inverse Gaussian (IG), and exponential lifetime-truncated distributions, a Shewhart-type attribute control chart is built to display the data. The performance of attributed control chart constructed on truncated life test is evaluated by average run length, which compares the performance of all distributions. Our study arranges that IG is better distribution among all.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

On a class of bivariate mixed Sarmanov distributions

Multivariate distributions are more and more used to model the dependence encountered in many fields. However, classical multivariate distributions can be restrictive by their nature, while Sarmanov's multivariate distribution, by joining different marginals in a flexible and tractable dependence structure, often provides a valuable alternative. In this paper, we introduce some bivariate mixed Sarmanov distributions with the purpose to extend the class of bivariate Sarmanov distributions and to obtain new dependency structures. Special attention is paid to the bivariate mixed Sarmanov distribution with Poisson marginals and, in particular, to the resulting bivariate Sarmanov distributions with negative binomial and with Poisson‐inverse Gaussian marginals; these particular types of mixed distributions have possible applications in, for example modelling bivariate count data. The extension to higher dimensions is also discussed. Moreover, concerning the dependency structure, we also present some correlation formulas.     

The optimal group size using inverse binomial group testing considering misclassification

ABSTRACT

Inverse binomial sampling is preferred for quick report. It is also recommended when the population proportion is really small to ensure a positive sample is contained. Group testing has been discussed extensively under binomial model, but not so much under negative binomial model. In this study, we investigate the problem of how to determine the group size using inverse binomial group testing. We propose to choose the optimal group size by minimizing asymptotic variance of the estimator or the cost relative to Fisher information. We show the good performance of our estimator by applying to the data of Chlamydia.     

Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its Derivatives

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.     

On a characterization of the dispersion matrix based on the properties of regression

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

Characterizations of gamma,inverse Gaussian,and negative binomial distributions via their length-biased distributions

A general form for characterizing inverse Gaussian and Wald distributions, based on their respective length-biased distributions, is introduced. Further results for characterizations of the gamma distribution, the negative binomial distribution and some mixtures of them by using their lengthbiased distributions are establised.     

An Extended Binomial Distribution with Applications

Based on Skellam (Poisson difference) distribution, an extended binomial distribution is introduced as a byproduct of extending Moran's characterization of Poisson distribution to the Skellam distribution. Basic properties of the distribution are investigated. Also, estimation of the distribution parameters is obtained. Applications with real data are also described.